Gain Compensated Directional Propagation Measurements

ABSTRACT

A method for obtaining gain compensated propagation measurements includes rotating an electromagnetic logging while drilling tool having at least one axial transmitter antenna, at least one transverse transmitter antenna, at least one axial receiver antenna and at least one transverse receiver antenna in a subterranean wellbore. Electromagnetic voltage measurements are acquired from the axial and transverse receiver antennas while rotating. The acquired voltage measurements are processed to compute harmonic voltage coefficients. Ratios of selected ones of the harmonic voltage coefficients are in turn processed to compute at least one gain compensated quantity including an axial cross term.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of the following four U.S.Provisional patent applications (each of which was filed on Mar. 29,2014): (i) Ser. No. 61/972,287 entitled Fully Gain Compensated TensorPropagation Cross-Term Measurements with Orthogonal Antennas, (ii) Ser.No. 61/972,288 entitled Improved Symmetrized and Anti-symmetrizedMeasurements with Orthogonal Antennas, (iii) Ser. No. 61/972,289entitled Compensated Directional Measurements using Tilted Moments thatare Independent of Tilt Angle Parameter, and (iv) Ser. No. 61/972,290entitled Compensated Array Configurations with Orthogonal Antennas.

FIELD OF THE DISCLOSURE

Disclosed embodiments relate generally to downhole electromagneticlogging methods and more particularly to a logging tool and a method formaking gain compensated directional propagation measurements, such asphase shift and attenuation measurements, using orthogonal antennas.

BACKGROUND INFORMATION

The use of electromagnetic measurements in prior art downholeapplications, such as logging while drilling (LWD) and wireline loggingapplications is well known. Such techniques may be utilized to determinea subterranean formation resistivity, which, along with formationporosity measurements, is often used to indicate the presence ofhydrocarbons in the formation. Moreover, azimuthally sensitivedirectional resistivity measurements are commonly employed e.g., inpay-zone steering applications, to provide information upon whichsteering decisions may be made.

Downhole electromagnetic measurements are commonly inverted at thesurface using a formation model to obtain various formation parameters,for example, including vertical resistivity, horizontal resistivity,distance to a remote bed, resistivity of the remote bed, dip angle, andthe like. One challenge in utilizing directional electromagneticresistivity measurements, is obtaining a sufficient quantity of data toperform a reliable inversion. The actual formation structure isfrequently significantly more complex than the formation models used inthe inversion. The use of a three-dimensional matrix of propagationmeasurements may enable a full three-dimensional measurement of theformation properties to be obtained as well as improve formation imagingand electromagnetic look ahead measurements. However, there are no knownmethods for providing a fully gain compensated tri-axial propagationmeasurement.

SUMMARY

A method for obtaining gain compensated electromagnetic logging whiledrilling propagation measurements is disclosed. An electromagneticlogging while drilling tool having at least one axial transmitterantenna, at least one transverse transmitter antenna at least one axialreceiver antenna, and at least one transverse receiver antenna isrotated in a subterranean wellbore. Electromagnetic voltage measurementsare acquired from the axial and transverse receiver antennas whilerotating. The acquired voltage measurements are processed to computeharmonic voltage coefficients. Ratios of selected ones of the harmonicvoltage coefficients are in turn processed to compute at least one gaincompensated quantity including an axial cross term.

The disclosed methodology provides a method for obtaining a gaincompensated three-dimensional matrix of measurements using orthogonalantennas. The acquired measurements are fully gain compensated andindependent of antenna tilt angle variation. Moreover, the disclosedmethod and apparatus tends to be insensitive to bending and alignmentangle errors.

This summary is provided to introduce a selection of concepts that arefurther described below in the detailed description. This summary is notintended to identify key or essential features of the claimed subjectmatter, nor is it intended to be used as an aid in limiting the scope ofthe claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the disclosed subject matter, andadvantages thereof, reference is now made to the following descriptionstaken in conjunction with the accompanying drawings, in which:

FIG. 1 depicts one example of a drilling rig on which the disclosedelectromagnetic logging methods may be utilized.

FIG. 2A depicts one example of the electromagnetic logging tool shown onFIG. 1.

FIG. 2B schematically depicts the antenna moments in an electromagneticlogging tool including triaxial transmitters and receivers.

FIGS. 3A, 3B, 3C, and 3D (collectively FIG. 3) depict the antennamoments for various example transmitter and receiver configurations forobtaining gain compensated axial cross term quantities.

FIG. 4 depicts a flow chart of one disclosed method embodiment forobtaining gain compensated axial cross term quantities.

FIGS. 5A and 5B (collectively FIG. 5) depict the antenna moments forvarious example transmitter and receiver configurations for obtaininggain compensated symmetrized and anti-symmetrized quantities.

FIG. 6 depicts a flow chart of one disclosed method embodiment forobtaining gain compensated symmetrized and anti-symmetrized quantities.

FIG. 7 depicts a three layer formation model used to evaluate thedirectional response of disclosed symmetrized and anti-symmetrizedmeasurements.

FIGS. 8A and 8B depict symmetrized and anti-symmetrized phase shift andattenuation versus total vertical depth at 30 degrees relative dip.

FIGS. 9A and 9B depict symmetrized and anti-symmetrized phase shift andattenuation versus total vertical depth at 70 degrees relative dip.

FIGS. 10A and 10B depict symmetrized and anti-symmetrized phase shiftand attenuation versus total vertical depth at 88 degrees relative dip.

FIGS. 11A, 11B, 11C, and 11D (collectively FIG. 11) depict the antennamoments for various example transmitter and receiver configurations forobtaining gain compensated transverse coupling and cross-couplingquantities.

FIG. 12 depicts a flow chart of one disclosed method embodiment 140 forobtaining gain compensated transverse term quantities.

FIG. 13 depicts the antenna moments for an electromagnetic logging toolincluding tilted transmitters.

DETAILED DESCRIPTION

FIG. 1 depicts an example drilling rig 10 suitable for employing variousmethod embodiments disclosed herein. A semisubmersible drilling platform12 is positioned over an oil or gas formation (not shown) disposed belowthe sea floor 16. A subsea conduit 18 extends from deck 20 of platform12 to a wellhead installation 22. The platform may include a derrick anda hoisting apparatus for raising and lowering a drill string 30, which,as shown, extends into borehole 40 and includes a drill bit 32 deployedat the lower end of a bottom hole assembly (BHA) that further includesan electromagnetic measurement tool 50 configured to make directionalelectromagnetic logging measurements. As described in more detail belowthe electromagnetic measurement tool 50 may include multiple orthogonalantennas deployed on a logging while drilling tool body.

It will be understood that the deployment illustrated on FIG. 1 ismerely an example. Drill string 30 may include substantially anysuitable downhole tool components, for example, including a steeringtool such as a rotary steerable tool, a downhole telemetry system, andone or more MWD or LWD tools including various sensors for sensingdownhole characteristics of the borehole and the surrounding formation.The disclosed embodiments are by no means limited to any particulardrill string configuration.

It will be further understood that the disclosed embodiments are notlimited to use with a semisubmersible platform 12 as illustrated onFIG. 1. The disclosed embodiments are equally well suited for use witheither onshore or offshore subterranean operations.

FIG. 2A depicts one example of an electromagnetic measurement tool 50.In the depicted embodiment measurement tool 50 includes first and secondaxially spaced transmitters 52 and 54 and first and second axiallyspaced receivers 56 and 58 deployed on a logging while drilling toolbody 51, with the receivers 56 and 58 being deployed axially between thetransmitters 52 and 54. As described in more detail below, each of thetransmitters 52 and 54 and receivers 56 and 58 includes at least onetransverse antenna and may further include an axial antenna. Forexample, the transmitters and receivers may include a bi-axial antennaarrangement including an axial antenna and a transverse (cross-axial)antenna. In another embodiment, the transmitters and receivers mayinclude a tri-axial antenna arrangement including an axial antenna andfirst and second transverse antennas that are orthogonal to one another.As is known to those of ordinary skill in the art, an axial antenna isone whose moment is substantially parallel with the longitudinal axis ofthe tool. Axial antennas are commonly wound about the circumference ofthe logging tool such that the plane of the antenna is substantiallyorthogonal to the tool axis. A transverse antenna is one whose moment issubstantially perpendicular to the longitudinal axis of the tool. Atransverse antenna may include, for example, a saddle coil (e.g., asdisclosed in U.S. Patent Publications 2011/0074427 and 2011/0238312 eachof which is incorporated by reference herein).

FIG. 2B depicts the moments (magnetic dipoles) of one embodiment ofmeasurement tool 50 in which the transmitters 52, 54 and receivers 56,58 each include a tri-axial antenna arrangement. Each of thetransmitters 52, 54 includes an axial antenna T1 _(z) and T2 _(z) andfirst and second transverse antennas T1 _(x), T1 _(y) and T2 _(x), T2_(y). Likewise, each of the receivers 56, 58 includes an axial antennaR1, and R2, and first and second transverse antennas R1 _(x), R1 _(y)and R2 _(x), R2 _(y). It will be understood that the disclosedembodiments are not limited to a tri-axial antenna configuration such asthat depicted on FIG. 2B.

As is known to those of ordinary skill in the art, a time varyingelectric current (an alternating current) in a transmitting antennaproduces a corresponding time varying magnetic field in the localenvironment (e.g., the tool collar and the formation). The magneticfield in turn induces electrical currents (eddy currents) in theconductive formation. These eddy currents further produce secondarymagnetic fields which may produce a voltage response in a receivingantenna. The measured voltage in the receiving antennae can beprocessed, as is known to those of ordinary skill in the art, to obtainone or more properties of the formation.

In general the earth is anisotropic such that its electrical propertiesmay be expressed as a three-dimensional tensor that contains informationon formation resistivity anisotropy, dip, bed boundaries and otheraspects of formation geometry. It will be understood by those ofordinary skill in the art that the mutual couplings between thetri-axial transmitter antennas and the tri-axial receiver antennasdepicted on FIG. 2B form a three-dimensional matrix and thus may havesensitivity to a full three-dimensional formation impedance tensor. Forexample, a three-dimensional matrix of measured voltages V may beexpressed as follows:

$\begin{matrix}{V_{ij} = {\begin{bmatrix}V_{ijxx} & V_{ijxy} & V_{ijxz} \\V_{ijyx} & V_{ijyy} & V_{ijyz} \\V_{ijzx} & V_{ijzy} & V_{ijzz}\end{bmatrix} = {{I_{i}Z_{ij}} = {\begin{bmatrix}I_{ix} & 0 & 0 \\0 & I_{iy} & 0 \\0 & 0 & I_{iz}\end{bmatrix}\begin{bmatrix}Z_{ijxx} & Z_{ijxy} & Z_{ijxz} \\Z_{ijyx} & Z_{ijyy} & Z_{ijyz} \\Z_{ijzx} & Z_{ijzy} & Z_{ijzz}\end{bmatrix}}}}} & (1)\end{matrix}$

where V_(ij) represent the three-dimensional matrix of measured voltageswith i indicating the corresponding transmitter triad (e.g., T1 or T2)and j indicating the corresponding receiver triad (e.g., R1 or R2),I_(i) represent the transmitter currents, and Z_(ij) represent thetransfer impedances which depend on the electrical and magneticproperties of the environment surrounding the antenna pair in additionto the frequency, geometry, and spacing of the antennas. The third andfourth subscripts indicate the axial orientation of the transmitter andreceiver antennas. For example, V_(12xy) represents a voltagemeasurement on the y-axis antenna of receiver R2 from a firing of thex-axis antenna of transmitter T1.

When bending of the measurement tool is negligible (e.g., less thanabout 10 degrees), the three dimensional voltage matrix may be modeledmathematically, for example, as follows:

V _(ij) =G _(Ti)(R _(θ) ^(t) Z _(ij) R _(θ))G _(Rj)  (2)

where Z_(ij) represent the transfer impedances as described above,G_(Ti) and G_(Rj) are diagonal matrices representing the transmitter andreceiver gains, R_(θ) represents the rotation matrix about the z-axisthrough angle θ, and the superscript t represents the transpose of thecorresponding matrix. The gain and rotation matrices in Equation 2 maybe given, for example, as follows:

$\begin{matrix}{G_{Ti} = \begin{bmatrix}g_{Tix} & 0 & 0 \\0 & g_{Tiy} & 0 \\0 & 0 & g_{Tiz}\end{bmatrix}} & (3) \\{G_{Rj} = \begin{bmatrix}g_{Rjx} & 0 & 0 \\0 & g_{Rjy} & 0 \\0 & 0 & g_{Rjz}\end{bmatrix}} & (4) \\{R_{\theta} = \begin{bmatrix}{\cos (\theta)} & {- {\sin (\theta)}} & 0 \\{\sin (\theta)} & {\cos (\theta)} & 0 \\0 & 0 & 1\end{bmatrix}} & (5)\end{matrix}$

The rotated couplings (shown in the parentheses in Equation 2) may beexpressed mathematically in harmonic form, for example, as follows:

R _(θ) ^(t) Z _(ij) R _(θ) =Z _(DC) _(—) _(ij) =Z _(FHC) _(—) _(ij)cos(θ)=Z _(FHS) _(—) _(ij) sin(θ)+Z _(SHC) _(—) _(ii) cos(2θ)+Z _(SHS)_(—) _(ij) sin(2θ)  (6)

where Z_(DC) _(—) _(ij) represents a DC (average) coupling coefficient,Z_(FHC) _(—) _(ij) and Z_(FHC) _(—) _(ij) represent first order harmoniccosine and first order harmonic sine coefficients, and Z_(SHC) _(—)_(ij) and Z_(SHS) _(—) _(ij) represent second order harmonic cosine andsecond order harmonic sine coefficients of the ij transmitter receivercouplings. These coefficients are shown below:

$Z_{{DC}_{—}{ij}} = \begin{bmatrix}\frac{Z_{ijxx} + Z_{ijyy}}{2} & \frac{\left( {Z_{ijxy} - Z_{ijyx}} \right)}{2} & 0 \\{- \frac{\left( {Z_{ijxy} - Z_{ijyx}} \right)}{2}} & \frac{Z_{ijxx} + Z_{ijyy}}{2} & 0 \\0 & 0 & Z_{ijzz}\end{bmatrix}$ $Z_{{FHC}_{—}{ij}} = \begin{bmatrix}0 & 0 & Z_{ijxz} \\0 & 0 & Z_{ijyx} \\Z_{ijzx} & Z_{ijzy} & 0\end{bmatrix}$ $Z_{{FHS}_{—}{ij}} = \begin{bmatrix}0 & 0 & Z_{ijxz} \\0 & 0 & {- Z_{ijxz}} \\Z_{ijzy} & {- Z_{ijzx}} & 0\end{bmatrix}$ $Z_{{SHC}_{—}{ij}} = \begin{bmatrix}\frac{Z_{ijxx} - Z_{ijyy}}{2} & \frac{\left( {Z_{ijxy} + Z_{ijyx}} \right)}{2} & 0 \\\frac{\left( {Z_{ijxy} + Z_{ijyx}} \right)}{2} & {- \frac{\left( {Z_{ijxx} - Z_{ijyy}} \right)}{2}} & 0 \\0 & 0 & 0\end{bmatrix}$ $\begin{matrix}{Z_{{SHS}_{ij}} = \begin{bmatrix}\frac{\left( {Z_{ijxy} + Z_{ijyx}} \right)}{2} & {- \frac{\left( {Z_{ijxx} - Z_{ijyy}} \right)}{2}} & 0 \\{- \frac{\left( {Z_{ijxx} - Z_{ijyy}} \right)}{2}} & {- \frac{\left( {Z_{ijxy} + Z_{ijyx}} \right)}{2}} & 0 \\0 & 0 & 0\end{bmatrix}} & (7)\end{matrix}$

In general, the receiving antenna voltages are measured while the toolrotates in the borehole. Following the form of Equation 6, the measuredvoltages may be expressed mathematically in terms of its harmonicvoltage coefficients, for example, as follows thereby enabling theharmonic voltage coefficients to be obtained:

V _(ij) =V _(DC) _(—) _(ij) +V _(FHC) _(—) _(ij) cos(θ)+V _(FHS) _(—)_(ij) sin(θ)+V _(SHC) _(—) _(ij) cos(2θ)+V _(SHS) _(—) _(ij)sin(2θ)  (8)

wherein where V_(DC) _(—) _(ij) represents a DC voltage coefficient,V_(FHC) _(—) _(ij) and V_(FHS) _(—) _(ij) represent first order harmoniccosine and first order harmonic sine voltage coefficients (also referredto herein as first harmonic cosine and first harmonic sine voltagecoefficients), and V_(SHC) _(—) _(ij) and V_(SHS) _(—) _(ij) representsecond order harmonic cosine and second order harmonic sine voltagecoefficients (also referred to herein as second harmonic cosine andsecond harmonic sine voltage coefficients) of the ij transmitterreceiver couplings.

Gain Compensated Axial Cross Terms

It will be understood that collocated tri-axial transmitter and receiverembodiments (e.g., as depicted on FIG. 2B) are not required to gaincompensate certain of the three-dimensional matrix components. Forexample, the axial cross terms (i.e., the xz, zx, yz, and zy terms) maybe gain compensated using any tool embodiment that includes an axialtransmitter antenna, a transverse (cross-axial) transmitter antenna, anaxial receiver antenna, and a transverse receiver antenna deployed onthe tool body. These transmitter and receiver antennas may bedistributed along the tool body with substantially any suitable spacingand order. Moreover, the transmitter antennas and/or the receiverantennas may be collocated. The disclosed embodiments are not limited toany particular transmitter and receiver antenna configuration so long asthe tool includes at least one axial transmitter antenna, at least onetransverse transmitter antenna, at least one axial receiver antenna, andat least one transverse receiver antenna.

FIGS. 3A, 3B, 3C, and 3D (collectively FIG. 3) depict the antennamoments for various example transmitter and receiver configurations forobtaining gain compensated axial cross terms (also referred to herein asaxial cross coupling impedances). FIG. 3A depicts an example toolembodiment 60 including a transmitter T axially spaced apart from areceiver R. The transmitter T includes collocated axial and transversetransmitting antennas having moments T_(z) and T_(x). The receiver Rincludes collocated axial and transverse receiving antennas havingmoments R_(z) and R_(x).

FIG. 3B depicts an alternative tool embodiment 65 that is similar totool embodiment 60 in that it also includes a transmitter T includingaxial and transverse transmitting antennas having moments T_(z) andT_(x). Tool embodiment 65 differs from tool embodiment 60 in that theaxial and transverse receiving antennas R_(z) and R_(x) are notcollocated, but are axially spaced apart from one another on the toolbody.

FIG. 3C depicts another alternative tool embodiment 70 that is similarto tool embodiment 60 in that it also includes a receiver R includingaxial and transverse receiving antennas having moments R_(z) and R_(x).Tool embodiment 70 differs from tool embodiment 60 in that the axial andtransverse transmitting antennas T_(z) and T_(x) are not collocated, butare axially spaced apart from one another on the tool body.

FIG. 3D depicts still another alternative tool embodiment 75 includingaxial and transverse transmitting antennas and axial and transversereceiving antennas T_(z) and T_(x) and R_(z) and R_(x). Tool embodiment75 differs from tool embodiment 60 in that neither the transmittingantennas nor the receiving antennas are collocated, but are axiallyspaced apart on the tool body. It will be understood that receiverantennas are not necessarily deployed between the transmitter antennasas depicted (TRRT), but may be axially distributed in substantially anyorder, for example, (i) with the transmitter antennas between thereceiver antennas (RTTR), (ii) with the transmitter antennas alternatingwith the receiver antennas (TRTR or RTRT), or (iii) with the transmitterantennas on one side and the receiver antennas on the other (TTRR orRRTT). It will thus be understood that the disclosed embodiments are notlimited to collocation or non-collocation of the axial and transversetransmitting and/or receiving antennas or to any particular spacing andlocation thereof along the tool body.

It will further be understood that one or more of the transmittersand/or receivers in tool embodiments 60, 65, 70, and 75 may optionallyfurther include a second transverse antenna such that the transmitterand/or receiver includes a triaxial antenna arrangement having threeantennas that are arranged to be mutually independent (e.g., as in FIG.2B).

FIG. 4 depicts a flow chart of one disclosed method embodiment 100 forobtaining one or more gain compensated axial cross terms. Anelectromagnetic measurement tool (e.g., one of the measurement toolsdepicted on FIG. 2B or FIG. 3) is rotated in a subterranean wellbore at102. Electromagnetic measurements are acquired at 104 while the tool isrotating and processed to obtain harmonic voltage coefficients. Ratiosof selected harmonic voltage coefficients may then be processed toobtain the gain compensated axial cross terms at 106.

The electromagnetic measurements may be acquired and processed to obtainharmonic voltage coefficients, for example, as describe above withrespect to Equations 1 through 8. As described above, gain compensatedaxial cross terms may be obtained using a measurement tool including anaxial transmitter antenna, a transverse transmitter antenna, an axialreceiver antenna, and a transverse receiver antenna (each of the toolembodiments depicted on FIGS. 2B and 3A-3D include such axial andtransverse transmitter and receiver antennas).

The measured voltages may be related to the impedances between thetransmitter and receiver antennas as described above. The DC, firstharmonic cosine, and first harmonic sine voltage coefficients may beexpressed, for example, as follows in terms of the couplings and therespective transmitter and receiver gains:

$V_{{DC}_{—}{xx}} = {g_{Tx}g_{Rx}\frac{Z_{xx} + Z_{yy}}{2}}$V_(DC_(—)zz) = g_(Tz)g_(Rz)Z_(zz)V_(FHC_(—)xz) = g_(Tx)g_(Rx)Z_(xz)V_(FHC_(—)zx) = g_(Tz)g_(Rx)Z_(zx)V_(FHS_(—)xz) = g_(Tx)g_(Rz)Z_(yz) $\begin{matrix}{V_{{FHS}_{—}{zx}} = {g_{Tz}g_{Rx}Z_{xy}}} & (9)\end{matrix}$

where g_(Tz) and g_(Tx) represent the gains of the axial and transversetransmitter antennas, g_(Rz) and g_(Rx) represent the gains of the axialand transverse receiver antennas, V_(DC) _(—) _(xx) is the DC voltageobtained from the x directed receiver when the x directed transmitterfires, V_(DC) _(—) _(zz) is the DC voltage obtained from the z directedreceiver when the z directed transmitter fires, V_(FHC) _(—) _(xz)(V_(FHS) _(—) _(xz)) is the first harmonic cosine (sine) voltageobtained from the z directed receiver when the x directed transmitterfires, and V_(FHC) _(—) _(zx) (V_(FHS) _(—) _(zx)) is the first harmoniccosine (sine) voltage obtained from the x directed receiver when the zdirected transmitter fires.

Selected ratios of the DC, first harmonic cosine, and first harmonicsine voltage coefficients given in Equation 9 may be processed at 106 tocompute the gain compensated axial cross terms. For example, a gaincompensated quantity (ratio) related to the xz and/or the zx crosscoupling impedances may be computed by processing a ratio of a productof the first harmonic cosine coefficients of the cross-coupling terms toa product of the DC coefficients of the direct coupling terms. Likewise,a gain compensated quantity (ratio) related to the yz and/or the zycross coupling impedances may be computed by processing a ratio of aproduct of the first harmonic sine coefficients of the cross-couplingterms to a product of the DC coefficients of the direct coupling terms.It will be understood that the xz, zx, yz, and zy cross couplingimpedances are also referred to herein as couplings. Such ratios may beexpressed mathematically in general terms (for example for theconfiguration shown on FIG. 3A) as follows:

${CRxz} = {\frac{V_{{FHC}_{—}{zx}} \cdot V_{{FHC}_{—}{xz}}}{V_{{DC}_{—}{xx}} \cdot V_{{DC}_{—}{zz}}} = \frac{2Z_{zx}Z_{xz}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}$$\begin{matrix}{{CRyz} = {\frac{V_{{FHS}_{—}{zx}} \cdot V_{{FHS}_{—}{xz}}}{V_{{DC}_{—}{xx}} \cdot V_{{DC}_{—}{zz}}} = \frac{2Z_{zy}Z_{yz}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}} & (10)\end{matrix}$

where CRxz and CRyz represent the gain compensated quantities (ratios).Note that the transmitter and receiver gains are fully canceled inEquation 10 resulting in the computed quantities being fully gaincompensated.

The following discussion makes use of the notation and the antennaspacing described above with respect to FIG. 2B, although it will beunderstood that the disclosed embodiments are not so limited. Asdescribed above, the axial cross term voltages (the xz, zx, yz, and zyterms) may be fully gain compensated using any tool embodiment thatincludes an axial transmitter antenna, a transverse transmitter antenna,an axial receiver antenna, and a transverse receiver antenna.

It will be understood that in general Z_(TR)=Z_(TR) ^(t). For example,the impedances Z_(12zx) and Z_(21xz) are identically equal in ahomogeneous anisotropic medium. These impedances are only approximatelyequal in a heterogeneous medium (e.g., in the presence of bedboundaries) since the transmitter-receiver pairs 12 zx and 21 xz are notexactly located at the same points in space. A gain compensated quantityCZX that has the characteristics of a zx tensor coupling element may beobtained, for example, as given in the following equations:

${CZX} = {\sqrt{\frac{V_{{FHC}_{—}12{zx}} \cdot V_{{FHC}_{—}21{xz}}}{V_{{DC}_{—}22{xx}} \cdot V_{{DC}_{—}11{zz}}}} \approx \sqrt{\frac{2Z_{zx}Z_{zy}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}$$\begin{matrix}{{CZX} = {\sqrt{\frac{V_{{FHC}_{—}11{zx}} \cdot V_{{FHC}_{—}22{xz}}}{V_{{DC}_{—}21{xx}} \cdot V_{{DC}_{—}12{zz}}}} \approx \sqrt{\frac{2Z_{zx}Z_{zx}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}} & (11)\end{matrix}$

Likewise a gain compensated quantity CZY that has the characteristics ofa zy tensor coupling element may be obtained, for example, as given inthe following equations:

${CZY} = {\sqrt{\frac{V_{{FHS}_{—}12{zx}} \cdot V_{{FHS}_{—}21{xz}}}{V_{{DC}_{—}22{xx}} \cdot V_{{DC}_{—}11{zz}}}} \approx \sqrt{\frac{2Z_{zy}Z_{zy}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}$$\begin{matrix}{{CZY} = {\sqrt{\frac{V_{{FHS}_{—}11{zx}} \cdot V_{{FHS}_{—}22{xz}}}{V_{{DC}_{—}21{xx}} \cdot V_{{DC}_{—}12{zz}}}} \approx \sqrt{\frac{2Z_{zy}Z_{zy}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}} & (12)\end{matrix}$

A gain compensated quantity CXZ that has the characteristics of a xztensor coupling element may be obtained, for example, as given in thefollowing equations:

${CXZ} = {\sqrt{\frac{V_{{FHC}_{—}21{zx}} \cdot V_{{FHC}_{—}12{xz}}}{V_{{DC}_{—}11{xx}} \cdot V_{{DC}_{—}22{zz}}}} \approx \sqrt{\frac{2Z_{xz}Z_{xz}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}$$\begin{matrix}{{CXZ} = {\sqrt{\frac{V_{{FHC}_{—}22{zx}} \cdot V_{{FHC}_{—}11{xz}}}{V_{{DC}_{—}12{xx}} \cdot V_{{DC}_{—}21{zz}}}} \approx \sqrt{\frac{2Z_{xz}Z_{xz}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}} & (13)\end{matrix}$

Likewise a gain compensated quantity CYZ that has the characteristics ofa yz tensor coupling element may be obtained, for example, as given inthe following equations:

${CYZ} = {\sqrt{\frac{V_{{FHS}_{—}21{zx}} \cdot V_{{FHS}_{—}12{xz}}}{V_{{DC}_{—}11{xx}} \cdot V_{{DC}_{—}22{zz}}}} \approx \sqrt{\frac{2Z_{yz}Z_{yz}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}$$\begin{matrix}{{CYZ} = {\sqrt{\frac{V_{{FHS}_{—}11{zx}} \cdot V_{{FHS}_{—}22{xz}}}{V_{{DC}_{—}12{xx}} \cdot V_{{DC}_{—}21{zz}}}} \approx \sqrt{\frac{2Z_{zy}Z_{zy}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}} & (14)\end{matrix}$

Gain compensated quantities may also be computed that are proportionalto a product of the xz and zx terms as well a product of the yz and zyterms. For example, continuing to make use of the notation and theantenna spacing described above with respect to FIG. 2B, the gaincompensated quantities CXZZX and CYZZY which are proportional to aproduct of the xz and zx terms and a product of the yz and zy terms maybe obtained as follows:

${CXZZX} = {\sqrt{\frac{V_{{FHC}_{—}{ijzx}} \cdot V_{{FHC}_{—}{ijxz}}}{V_{{DC}_{—}{ijxx}} \cdot V_{{DC}_{—}{ijzz}}}} \approx \sqrt{\frac{2Z_{xz}Z_{zx}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}$$\begin{matrix}{{CYZZY} = {\sqrt{\frac{V_{{FHS}_{—}{ijzx}} \cdot V_{{FHS}_{—}{ijxz}}}{V_{{DC}_{—}{ijxx}} \cdot V_{{DC}_{—}{ijzz}}}} \approx \sqrt{\frac{2Z_{yz}Z_{zy}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}} & (15)\end{matrix}$

Gain compensated quantities which are related to an xz zx product and ayz zy product may further be obtained as follows:

${CXZZX} = {\sqrt{\frac{V_{{FHC}_{—}{ijzx}} \cdot V_{{FHC}_{—}{iixz}}}{V_{{DC}_{—}{ijxx}} \cdot V_{{DC}_{—}{iizz}}}} \approx \sqrt{\frac{2Z_{xz}Z_{zx}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}$$\begin{matrix}{{{CYZZY} = {\sqrt{\frac{V_{{FHS}_{—}{ijzx}} \cdot V_{{FHS}_{—}{iixz}}}{V_{{DC}_{—}{ijxx}} \cdot V_{{DC}_{—}{iizz}}}} \approx \sqrt{\frac{2Z_{yz}Z_{zy}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}}{{and}\text{:}}{{CXZZX} = {\sqrt{\frac{V_{{FHC}_{—}{iizx}} \cdot V_{{FHC}_{—}{jixz}}}{V_{{DC}_{—}{jixx}} \cdot V_{{DC}_{—}{iizz}}}} \approx \sqrt{\frac{2Z_{xz}Z_{zx}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}}} & (16) \\{{CYZZY} = {\sqrt{\frac{V_{{FHS}_{—}{iizx}} \cdot V_{{FHS}_{—}{jixz}}}{V_{{DC}_{—}{jixx}} \cdot V_{{DC}_{—}{iizz}}}} \approx \sqrt{\frac{2Z_{yz}Z_{zy}}{Z_{zz}\left( {Z_{xx} + Z_{yy}} \right)}}}} & (17)\end{matrix}$

It will be understood that when using the notation and antenna spacingdescribed above with respect to FIG. 2B, (i) Equation 15 relates to atool embodiment including collocated axial and transverse transmitterantennas and collocated axial and transverse receiver antennas, (ii)Equation 16 relates to a tool embodiment including collocated axial andtransverse transmitter antennas and non-collocated axial and transversereceiver antennas, and (iii) Equation 17 relates to a tool embodimentincluding non-collocated axial and transverse transmitter antennas andcollocated axial and transverse receiver antennas.

Table 1 lists various antenna configurations from which gain compensatedaxial cross terms may be obtained. For example the top entry in Column 1has the z transmitter in the left-most position, a z receiver in thenext position, an x receiver, then an x transmitter in the right-mostposition. Using the (approximate) symmetry Z_(TR)=Z_(TR) ^(t), each ofthese configurations, may have the character of a tensor componentrelated to one of the following products: zx·zx, zx·xz, xz·zx, or xz·xz.Column 1 lists antenna combinations from which a gain compensatedquantity related to the zx·zx product may be obtained. Column 2 listsantenna combinations from which a gain compensated quantity related tothe xz·xz product may be obtained. Column 3 lists antenna combinationsfrom which a gain compensated quantity related to the xz·zx product maybe obtained. And column 4 lists antenna combinations from which a gaincompensated quantity related to zx·xz product may be obtained. As notedabove, gain compensated axial cross terms may be obtained using any toolconfiguration including an axial transmitter antenna, a transverse(cross-axial) transmitter antenna, an axial receiver antenna, and atransverse receiver antenna.

In Table 1 the transmitter and receiver antennas are listed from upholeto downhole positions on the tool (from left to right in the table). Forexample, T_(z) R_(z) R_(x) T_(x) indicates an axial transmitter antennaT_(z) located above an axial receiver antenna R_(z) and a transversereceiver antenna R_(x) located above a transverse transmitter antennaT_(x). Adjacent transmitter antennas or receiver antennas may becollocated or non-collocated. In the above example, the axial receiverantenna R_(z) and the transverse receiver antenna R_(x) may becollocated or non-collocated such that the axial receiver antenna isabove the transverse receiver antenna.

TABLE 1 Compensated Compensated Compensated ZX XZ Compensated XZ · ZX ZX· XZ T_(z) R_(z) R_(x) T_(x) R_(x) T_(z) T_(x) R_(z) R_(z) R_(x) T_(z)T_(x) T_(z) R_(x) T_(x) R_(z) T_(z) R_(z) T_(x) R_(x) R_(x) T_(x) T_(z)R_(z) R_(z) R_(x) T_(x) T_(z) T_(z) T_(x) R_(z) R_(x) T_(z) R_(x) R_(z)T_(x) R_(x) T_(x) R_(z) T_(z) R_(z) T_(x) R_(x) T_(z) T_(z) T_(x) R_(x)R_(z) R_(z) T_(z) R_(x) T_(x) T_(x) R_(z) R_(x) T_(z) R_(x) T_(z) R_(z)T_(x) T_(x) T_(z) R_(z) R_(x) R_(z) T_(z) T_(x) R_(x) T_(x) R_(x) T_(z)R_(z) R_(x) R_(z) T_(z) T_(x) T_(x) T_(z) R_(x) R_(z) R_(z) T_(x) T_(z)R_(x) T_(x) R_(x) R_(z) T_(z) R_(x) R_(z) T_(x) T_(z) T_(x) R_(z) T_(z)R_(x)

It will be understood that since computation of the compensatedquantities in Equations 11-17 involves taking a square root, there maybe a 180 degree phase ambiguity (i.e., a sign ambiguity). As such, thegain ratios of the receivers may not be arbitrary, but should becontrolled such that they are less than 180 degrees (i.e., the antennawires should be connected to the electronics in the same way). Forun-tuned receiving antennas, the electronic and antenna gain/phasemismatch (assuming the antenna wires are not flipped from one receiverto another) may generally be controlled to within about 30 degrees(particularly at the lower frequencies used for deep measurements). Thisis well within 180 degrees (even at elevated temperatures where themismatch may be at its greatest).

A phase shift and attenuation may be computed for the compensatedquantities listed above, for example, as follows:

${PS}\mspace{14mu} \overset{def}{=}\mspace{14mu} {\frac{180}{\pi}{{angle}\left( {1 + {CQ}} \right)}}$$\begin{matrix}{{AT}\mspace{14mu} \overset{def}{=}\mspace{11mu} {20\; \log \; 10\left( {1 + {CQ}} \right)}} & (18)\end{matrix}$

where PS represents the phase shift, AT represents attenuation, and CQrepresents the compensated quantity (e.g., one of the quantitiescomputed in Equations 11-17). These quantities may be equal to zero insimple formations. Thus, the phase shift and attenuation were computedby adding one to CQ in Equation 18.

Gain Compensated Symmetrized and Anti-Symmetrized Quantities

Symmetrized and anti-symmetrized directional resistivity quantities havebeen disclosed in U.S. Pat. Nos. 6,969,994 and 7,536,261 which areincorporated by reference herein in their entireties. In general, thesymmetrized quantity is taken to be proportional to a difference betweenthe xz and zx terms while the anti-symmetrized quantity is taken to beproportional to a sum of the xz and zx terms. The symmetrizedmeasurement tends to be sensitive to bed boundaries and less sensitiveto anisotropy and dip while the anti-symmetrized measurement tends to besensitive to anisotropy and dip and less sensitive to bed boundaries.

It will be understood that a tool configuration including collocatedtri-axial transmitter and receiver embodiments (e.g., as depicted onFIG. 2B) is not required to obtain gain compensated symmetrized andanti-symmetrized quantities. Logging tool embodiments similar to thosedescribed above for obtaining gain compensated axial cross terms may beutilized. For example, FIGS. 5A and 5B (collectively FIG. 5) depictantenna moments for various example transmitter and receiverconfigurations for obtaining gain compensated symmetrized andanti-symmetrized quantities. FIG. 5A depicts an embodiment including anaxial receiver antenna R_(z) and a transverse receiver antenna R_(x)deployed axially between first and second transmitters T1 and T2. Eachof the transmitters includes an axial transmitter antenna T1 _(z) and T2_(z) and a transverse transmitter antenna T1 _(x) and T2 _(x). FIG. 5Bdepicts an embodiment including an axial transmitter antenna T_(z) and atransverse transmitter antenna T_(x) deployed axially between first andsecond receivers R1 and R2. Each of the receivers includes an axialreceiver antenna R1 _(z) and R2 _(z) and a transverse receiver antennaR1 _(x) and R2 _(x). While the embodiments depicted on FIG. 5 includenon-collocated antennas it will be understood that the axial andtransverse transmitter and receiver antennas may optionally becollocated.

FIG. 6 depicts a flow chart of one disclosed method embodiment 120 forobtaining gain compensated symmetrized and anti-symmetrized quantities.An electromagnetic measurement tool (e.g., one of the measurement toolsdepicted on FIG. 2B or FIG. 5) is rotated in a subterranean wellbore at122. Electromagnetic measurements are acquired at 124 while the tool isrotating and processed to obtain harmonic voltage coefficients. Ratiosof selected harmonic voltage coefficients may then be processed toobtain the gain compensated symmetrized and anti-symmetrized quantitiesat 126. It will be understood that the harmonic voltage coefficients maybe rotated to substantially any suitable reference angle prior tocomputing the gain compensated quantities at 126.

A gain compensated symmetrized measurement may be obtained, for example,via subtracting CZX from CXZ (e.g., as given in Equations 11 and 13) orvia subtracting CZY from CYZ (e.g., as given in Equation 12 and 14).Likewise a gain compensated anti-symmetrized measurement may beobtained, for example, via adding CZX to CXZ (e.g., as given inEquations 11 and 13) or via adding CZY to CYZ (e.g., as given inEquation 12 and 14).

Symmetrized and anti-symmetrized coupling quantities S and A may furtherbe expressed as combinations of products of the cross terms, forexample, as follows:

S=√{square root over (Z _(xz) ² +Z _(zx) ²×2Z _(xz) Z _(zx))}

A=√{square root over (Z _(xz) ² +Z _(zx) ²+2Z _(xz) Z _(zx))}  (19)

Recognizing that CZX is proportional to Z_(zx) (from Equation 11), CXZis proportional to Z_(xz) (from Equation 13), and CXZZX is proportionalto the square root of Z_(xz)·Z_(zx), gain compensated symmetrized Sc andanti-symmetrized Ac quantities may be given, for example, as follows:

Sc=√{square root over (CXZ ² +CZX ²−2CXZZX ²)}

Ac=√{square root over (CXZ ² +CZX ²+2CXZZX ²)}  (20)

where CXZ, CZX, and CXZZX may be obtained for example as described abovewith respect to Equations 10 through 17. Equation 20 may optionallyfurther include a scaling factor to ensure that Sc is equal to zero in ahomogeneous anisoptropic medium.

Following the notation and antenna spacing described above with respectto FIG. 2B, one particular embodiment of the symmetrized andanti-symmetrized quantities may be obtained by taking the followingratios:

$R_{zx} = {\frac{V_{{FHC}_{—}12{zx}} \cdot V_{{FHC}_{—}21{xz}}}{V_{{DC}_{—}22{xx}} \cdot V_{{DC}_{—}11{zz}}} \sim Z_{zx}^{2}}$$R_{xz} = {\frac{V_{{FHC}_{—}12{xz}} \cdot V_{{FHC}_{—}21{zx}}}{V_{{DC}_{—}11{xx}} \cdot V_{{DC}_{—}22{zz}}} \sim Z_{xz}^{2}}$${R\; 1_{xzzx}} = {\frac{V_{{FHC}_{—}12{zx}} \cdot V_{{FHC}_{—}12{xz}}}{V_{{DC}_{—}12{xx}} \cdot V_{{DC}_{—}12{zz}}} \sim {Z_{xz}Z_{zx}}}$$\begin{matrix}{{R\; 2_{xzzx}} = {\frac{V_{{FHC}_{—}21{zx}} \cdot V_{{FHC}_{—}21{xz}}}{V_{{DC}_{—}21{xx}} \cdot V_{{DC}_{—}21{zz}}} \sim {Z_{xz}Z_{zx}}}} & (21)\end{matrix}$

It will be readily apparent that the ratios in Equation 21 are fullygain compensated and similar to the gain compensated quantitiespresented above with respect to Equations 11-17. It will be understoodthat corresponding ratios R_(zy), R_(yz), R1 _(yzzy), and R2 _(yzzy) maybe computed by replacing the first harmonic cosine coefficients withcorresponding first harmonic sine coefficients. These ratios may beequivalently utilized to obtain the symmetrized and anti-symmetrizedquantities.

To combine the quantities in Equation 21 such that the symmetric resultis zero in a homogeneous anisotropic formation may require a scalingfactor. Such a scaling factor may be obtained, for example, as follows:

$\begin{matrix}{{scale} = \frac{V_{{DC}_{—}12{zz}}V_{{DC}_{—}21{zz}}V_{{DC}_{—}12{xx}}V_{{DC}_{—}21{xx}}}{V_{{DC}_{—}11{zz}}V_{{DC}_{—}22{zz}}V_{{DC}_{—}11{xx}}V_{{DC}_{—}22{xx}}}} & (22)\end{matrix}$

such that the fully gain compensated symmetrized and anti-symmetrizedquantities may be expressed as follows:

Sc=√{square root over (R _(xz) +R _(zx)−scale(R1_(xzzx) +R2_(xzzx)))}

Ac=√{square root over (R _(xz) +R _(zx)+scale(R1_(xzzx)+R2_(xzzx)))}  (23)

As described above with respect to Equations 11-17, taking the squareroot of a quantity can introduce a sign (or phase) ambiguity. Even withcareful unwrapping of the phase in Equation 23, a symmetrizeddirectional measurement Sc may have the same sign whether an approachingbed is above or below the measurement tool. The correct sign may beselected, for example, via selecting the sign of the phase orattenuation of the following relation:

TSD=√{square root over (R _(zx))}−√{square root over (R _(xz))}  (24)

where R_(zx) and R_(xz) are given in Equation 21. Similarly theanti-symmetrized directional measurement Ac in Equation 23 has the samesign whether the dip azimuth of the anisotropy is less than 180 degreesor greater than 180 degrees. This sign ambiguity may be resolved, forexample, by taking the sign of the phase or attenuation of the followingrelation.

TAD=√{square root over (R _(zx))}+√{square root over (R _(xz))}  (25)

The symmetrized and anti-symmetrized measurements may now be re-defined,for example, as follows to eliminate the sign ambiguity.

Sc

2sign(angle(TSD))√{square root over (R _(zx) +R _(xz)−scale(R1_(xz) _(—)_(zx) +R2_(xz) _(—) _(zx)))}

Ac

2sign(angle(TAD))√{square root over (R _(zx) +R _(xz)+scale(R1_(xz) _(—)_(zx) +R2_(xz) _(—) _(zx)))}  (26)

Symmetrized directional phase shift and attenuation measurements TDSPand TDSA may then be defined, for example, as follows:

${TDSP}\mspace{14mu} \overset{def}{=}\mspace{14mu} {\frac{180}{\pi}{{angle}\left( {1 + {Sc}} \right)}}$$\begin{matrix}{{TDSA}\mspace{14mu} \overset{def}{=}\mspace{14mu} {20\log \; 10\left( {1 + {Sc}} \right)}} & (27)\end{matrix}$

Likewise, anti-symmetrized directional phase shift and attenuation TDAPand TDAA measurements may also be defined, for example, as follows:

${TDAP}\mspace{14mu} \overset{def}{=}\mspace{14mu} {\frac{180}{\pi}{{angle}\left( {1 + {Ac}} \right)}}$$\begin{matrix}{{TDAA}\mspace{14mu} \overset{def}{=}\mspace{14mu} {20\log \; 10\left( {1 + {Ac}} \right)}} & (28)\end{matrix}$

The disclosed embodiments are now described in further detail withrespect to the following non-limiting examples in FIGS. 7, 8A, 8B, 9A,9B, 10A, and 10B. These examples are analytical (mathematical) and werecomputed using Equations 26-28 via software code developed using a pointdipole model.

FIG. 7 depicts a three layer formation model used to evaluate theresponse of the compensated symmetrized and anti-symmetrized measurementquantities described above with respect to Equations 27, 28, and 29. Theupper layer had a horizontal resistivity of 2 ohm·m and a verticalresistivity of 5 ohm·m. The middle layer had a horizontal and verticalresistivities of 200 ohm·m while the lower layer had a horizontalresistivity of 5 ohm·m and a vertical resistivity of 10 ohm·m. The upperand lower boundaries of the middle layer were at −15 and +15 feet,respectively. The electromagnetic tool was inclined at a non-zero dipangle D. In the examples that follow (in FIGS. 8A though 10B), a toolmodel configuration similar to that shown on FIG. 2B was utilized. Thereceiver R1 and transmitter T1 were located at +13 and +40 inches withrespect to the midpoint between receivers R1 and R2. The receiver R2 andthe transmitter T2 were located at −13 and −40 inches. Zero depth wasdefined as the depth at which the midpoint between receivers R1 and R2crossed the midpoint of the middle layer in the formation on FIG. 7.

FIGS. 8A and 8B depict symmetrized and anti-symmetrized phase shift andattenuation versus total vertical depth at 30 degrees relative dip.FIGS. 9A and 9B depict symmetrized and anti-symmetrized phase shift andattenuation versus total vertical depth at 70 degrees relative dip.FIGS. 10A and 10B depict symmetrized and anti-symmetrized phase shiftand attenuation versus total vertical depth at 88 degrees relative dip.These figures illustrate that the symmetrized quantity is zero away fromthe bed boundary. Near the boundary, the sign changes depending onwhether the bed is approached from above or below. The magnitude of thesymmetrized response is substantially independent of anisotropy. Theanti-symmetrized quantity is sensitive to anisotropy and dip withcomparatively less sensitivity to the bed boundaries.

Gain Compensated Transverse Terms

It will be understood that collocated tri-axial transmitter and receiverembodiments (e.g., as depicted on FIG. 2B) are not required to obtaingain compensated transverse terms (i.e., the xx and yy direct couplingimpedances and the xy and yx cross coupling impedances). The xx and yydirect coupling impedances and the xy and yx cross coupling impedancesare also referred to herein as xx, yy, xy, and yx couplings. These termsmay be gain compensated, for example, using any tool embodiment thatincludes x- and y-axis transmitter antennas and x- and y-axis receiverantennas deployed on the tool body. These transmitter and receiverantennas may be distributed along the tool body with substantially anysuitable spacing and order. Moreover, the transmitter antennas and/orthe receiver antennas may optionally be collocated. The disclosedembodiments are not limited to any particular transmitter and receiverantenna configuration so long as the tool includes at least x- andy-axis transmitter antennas and x- and y-axis receiver antennas.

FIGS. 11A and 11B depict the antenna moments for various exampletransmitter and receiver configurations suitable for obtaining gaincompensated xy and yx couplings. FIG. 11A depicts an example toolembodiment 80 including a transmitter T axially spaced apart from areceiver R. The transmitter T includes collocated x- and y-axistransmitting antennas having moments T_(x) and T_(y). The receiver Rincludes collocated x- and y-axis receiving antennas having momentsR_(x) and R_(y).

FIG. 11B depicts an alternative tool embodiment 85 including x- andy-axis transmitter antennas and x- and y-axis receiver antennas T_(x)and T_(y) and R_(x) and R_(y). Tool embodiment 85 differs from toolembodiment 80 in that neither the transmitter antennas nor the receiverantennas are collocated, but are axially spaced apart on the tool body.It will be understood that the receiver antennas are not necessarilydeployed between the transmitter antennas as depicted (TRRT), but may beaxially distributed in substantially any order, for example, (i) withthe transmitter antennas between the receiver antennas (RTTR), (ii) withthe transmitter antennas alternating with the receiver antennas (TRTR orRTRT), or (iii) with the transmitter antennas on one side and thereceiver antennas on the other (TTRR or RRTT). It will thus beunderstood that the disclosed embodiments are not limited to collocationor non-collocation of the axial and transverse transmitting and/orreceiving antennas or to any particular spacing or location thereofalong the tool body.

FIGS. 11C and 11D depict the antenna moments for various exampletransmitter and receiver configurations suitable for obtaining gaincompensated xx and yy couplings and xy and yx couplings. FIG. 11Cdepicts another tool embodiment 90 that is similar to tool embodiment 50shown on FIG. 2B in that it includes first and second transmitters T1and T2 and first and second receivers R1 and R2. Tool embodiment 90includes collocated x- and y-axis transmitters T1 _(x) and T1 _(y) andcollocated x- and y-axis receivers R2 _(x) and R2 _(y). Tool embodiment80 further includes x-axis receiver R1 _(x) and x-axis transmitter T2_(x). FIG. 11D depicts still another alternative tool embodiment 95including x-axis and y-axis transmitters and receivers. Tool embodiment95 includes first and second x-axis receivers R1 _(x) and R2 _(x)deployed axially between first and second x-axis transmitters T1 _(x)and T2 _(x). Tool embodiment 95 further includes a y-axis receiver R3_(y) deployed between the x-axis receivers and a y-axis transmitter T4_(y).

With continued reference to FIG. 11, it will be understood that one ormore or all of the transmitters and/or receivers depicted in toolembodiments 80, 85, 85, and 95 may further include axial (z-axis)antennas such that the transmitter and/or receiver includes a crossaxial pair of antennas or a triaxial antenna arrangement. Moreover, oneor more axial antennas may be located substantially anywhere in thedepicted antenna arrays. The disclosed embodiments are not limited inregard to the inclusion or location of axial antennas.

FIG. 12 depicts a flow chart of one disclosed method embodiment 140 forobtaining gain compensated transverse term quantities. Anelectromagnetic measurement tool (e.g., one of the measurement toolsdepicted on FIG. 2B or FIG. 11) is rotated in a subterranean wellbore at142. Electromagnetic measurements are acquired at 144 while the tool isrotating and processed to obtain harmonic voltage coefficients. Ratiosof selected harmonic voltage coefficients may then be processed toobtain the gain compensated transverse terms (the xx and/or yy couplingsand the xy and/or yx couplings).

The electromagnetic measurements may be acquired and processed to obtainharmonic coefficients, for example, as describe above with respect toEquations 1 through 8. Following Equations 3, 4, 7, and 8 and withrespect to FIG. 2B, the DC and second harmonic voltages may beexpressed, for example, as follows in terms of the couplings and therespective transmitter and receiver gains.

$V_{{DC}_{—}{xx}} = {g_{Tx}g_{Rx}\frac{Z_{xx} + Z_{yy}}{2}}$$V_{{DC}_{—}{xy}} = {g_{Tx}g_{Ry}\frac{Z_{xy} - Z_{yx}}{2}}$$V_{{DC}_{—}{yx}} = {g_{Ty}g_{Rx}\frac{Z_{yx} - Z_{xy}}{2}}$$V_{{DC}_{—}{yy}} = {g_{Ty}g_{Ry}\frac{Z_{xx} + Z_{yy}}{2}}$$V_{{SHC}_{—}{xx}} = {g_{Tx}g_{Rx}\frac{Z_{xx} - Z_{yy}}{2}}$$V_{{SHC}_{—}{xy}} = {g_{Tx}g_{Ry}\frac{Z_{xy} + Z_{yx}}{2}}$$V_{{SHC}_{—}{yx}} = {g_{Ty}g_{Rx}\frac{Z_{yx} + Z_{xy}}{2}}$$V_{{SHC}_{—}{yy}} = {g_{Ty}g_{Ry}\frac{Z_{yy} - Z_{xx}}{2}}$$V_{{SHS}_{—}{xx}} = {g_{Tx}g_{Rx}\frac{Z_{xy} + Z_{yx}}{2}}$$V_{{SHS}_{—}{xy}} = {g_{Tx}g_{Ry}\frac{Z_{yy} - Z_{xx}}{2}}$$V_{{SHS}_{—}{yx}} = {g_{Ty}g_{Rx}\frac{Z_{yy} - Z_{xx}}{2}}$$\begin{matrix}{V_{{SHS}_{—}{yy}} = {g_{Ty}g_{Ry}\frac{{- Z_{xy}} - Z_{yx}}{2}}} & (29)\end{matrix}$

where g_(Tx) and g_(Ty) represent the gains of the x-axis and y-axistransmitter antennas and g_(Rx) and g_(Ry) represent the gains of thex-axis and y-axis receiver antennas.

Selected ratios of the DC and second harmonic voltages given in Equation29 may be processed at 146 to compute the gain compensated quantitiesincluding the transverse terms. For example, following the notation andantenna spacing described above with respect to FIG. 2B, a gaincompensated xx quantity may be computed from either the xx or yy voltagemeasurements as follows:

${CXX}_{xx} = {\sqrt{\frac{\left( {V_{{DC}_{—}12{xx}} + V_{{SHC}_{—}12{xx}}} \right)\left( {V_{{DC}_{—}21{xx}} + V_{{SHC}_{—}21{xx}}} \right)}{\left( {V_{{DC}_{—}11{xx}} + V_{{SHC}_{—}11{xx}}} \right)\left( {V_{{DC}_{—}22{xx}} + V_{{SHC}_{—}22{xx}}} \right)}} = \sqrt{\frac{Z_{12{xx}}Z_{21{xx}}}{Z_{11{xx}}Z_{11{xx}}}}}$$\begin{matrix}{{CXX}_{yy} = {\sqrt{\frac{\left( {V_{{DC}_{—}12{yy}} \pm V_{{SHC}_{—}12{yy}}} \right)\left( {V_{{DC}_{—}21{yy}} - V_{{SHC}_{—}21{yy}}} \right)}{\left( {V_{{DC}_{—}11{yy}} \pm V_{{SHC}_{—}11{yy}}} \right)\left( {V_{{DC}_{—}22{yy}} \pm V_{{SHC}_{—}22{yy}}} \right)}} = \sqrt{\frac{Z_{12{xx}}Z_{21{xx}}}{Z_{11{xx}}Z_{11{xx}}}}}} & (30)\end{matrix}$

where CXX_(xx) and CXX_(yy) represent the gain compensated xx quantitiescomputed from the xx and yy voltage measurements. Since these quantitiesare identical they may be combined (e.g., averaged) to improve thesignal to noise ratio.

A gain compensated yy quantity may also be computed from either the xxor yy voltage measurements as follows:

${CYY}_{xx} = {\sqrt{\frac{\left( {V_{{DC}_{—}12{xx}} - V_{{SHC}_{—}12{xx}}} \right)\left( {V_{{DC}_{—}21{xx}} - V_{{SHC}_{—}21{xx}}} \right)}{\left( {V_{{DC}_{—}11{xx}} - V_{{SHC}_{—}11{xx}}} \right)\left( {V_{{DC}_{—}22{xx}} - V_{{SHC}_{—}22{xx}}} \right)}} = \sqrt{\frac{Z_{12{yy}}Z_{21{yy}}}{Z_{11y}Z_{11{yy}}}}}$$\begin{matrix}{{CYY}_{yy} = {\sqrt{\frac{\left( {V_{{DC}_{—}12{yy}} \mp V_{{SHC}_{—}12{yy}}} \right)\left( {V_{{DC}_{—}21{yy}} \mp V_{{SHC}_{—}21{yy}}} \right)}{\left( {V_{{DC}_{—}11{yy}} \mp V_{{SHC}_{—}11{yy}}} \right)\left( {V_{{DC}_{—}22{yy}} \mp V_{{SHC}_{—}22{yy}}} \right)}} = \sqrt{\frac{Z_{12{yy}}Z_{21{yy}}}{Z_{11{yy}}Z_{11{yy}}}}}} & (31)\end{matrix}$

where CYY_(xx) and CYY_(yy) represent the gain compensated yy quantitiescomputed from the xx and yy voltage measurements. These quantities arealso identical and may be combined (e.g., averaged) to improve thesignal to noise ratio.

Various gain compensated quantities that are combinations of the xx andyy couplings may also be computed from the xx and/or yy voltagemeasurements. A compensated quantity proportional to the sum of the xxand yy couplings may be computed from the xx and/or yy voltagemeasurements, for example, as follows:

${CXXplusYY}_{xx} = {\sqrt{\frac{V_{{DC}_{—}12{xx}}V_{{DC}_{—}21{xx}}}{V_{{DC}_{—}11{xx}}V_{{DC}_{—}22{xx}}}} = \sqrt{\frac{\left( {Z_{12{xx}} + Z_{12{yy}}} \right)\left( {Z_{21{xx}} + Z_{21{yy}}} \right)}{\left( {Z_{11{xx}} + Z_{11{yy}}} \right)\left( {Z_{22{xx}} + Z_{22{yy}}} \right)}}}$$\begin{matrix}{{CXXplusYY}_{yy} = {\sqrt{\frac{V_{{DC}_{—}12{yy}}V_{{DC}_{—}21{yy}}}{V_{{DC}_{—}11{yy}}V_{{DC}_{—}22{yy}}}} = \sqrt{\frac{\left( {Z_{12{xx}} + Z_{12{yy}}} \right)\left( {Z_{21{xx}} + Z_{21{yy}}} \right)}{\left( {Z_{11{xx}} + Z_{11{yy}}} \right)\left( {Z_{22{xx}} + Z_{22{yy}}} \right)}}}} & (32)\end{matrix}$

where CXXplusYY_(xx) and CXXplusYY_(yy) represent gain compensatedquantities computed from the xx and yy voltage measurements. Acompensated quantity proportional to the difference between the xx andyy couplings may be computed from the xx and/or yy voltage measurements,for example, as follows:

${CXXminusYY}_{xx} = {\sqrt{\frac{V_{{SHC}_{—}12{xx}}V_{{SHC}_{—}21{xx}}}{V_{{DC}_{—}11{xx}}V_{{DC}_{—}22{xx}}}} = \sqrt{\frac{\left( {Z_{12{xx}} - Z_{12{yy}}} \right)\left( {Z_{21{xx}} - Z_{21{yy}}} \right)}{\left( {Z_{11{xx}} + Z_{11{yy}}} \right)\left( {Z_{22{xx}} + Z_{22{yy}}} \right)}}}$$\begin{matrix}{{CXXminusYY}_{yy} = {\sqrt{\frac{V_{{SHC}_{—}12{yy}}V_{{SHC}_{—}21{yy}}}{V_{{DC}_{—}11{yy}}V_{{DC}_{—}22{yy}}}} = \sqrt{\frac{\left( {Z_{12{xx}} - Z_{12{yy}}} \right)\left( {Z_{21{xx}} - Z_{21{yy}}} \right)}{\left( {Z_{11{xx}} + Z_{11{yy}}} \right)\left( {Z_{22{xx}} + Z_{22{yy}}} \right)}}}} & (33)\end{matrix}$

where CXXminusYY_(xx) and CXXminusYY_(yy) represent gain compensatedquantities computed from the xx and yy voltage measurements. Acompensated quantity proportional to the difference between the xx andyy components may also be computed from the xx and/or yy voltagemeasurements, for example, as follows:

${CXXminusYY}_{ijxx} = {\frac{V_{{SHC}_{—}{ijxx}}}{V_{{DC}_{ijxx}}} = \frac{\left( {Z_{ijxx} - Z_{ijyy}} \right)}{\left( {Z_{ijxx} + Z_{ijyy}} \right)}}$$\begin{matrix}{{CXXminusYY}_{ijyy} = {{- \frac{V_{{SHC}_{—}{ijyy}}}{V_{{DC}_{—}{ijyy}}}} = \frac{\left( {Z_{ijxx} - Z_{ijyy}} \right)}{\left( {Z_{ijxx} + Z_{ijyy}} \right)}}} & (34)\end{matrix}$

where CXXminusYY_(ijxx) and CXXminusYY_(ijyy) represent gain compensatedquantities computed from the xx and yy voltage measurements.

Gain compensated quantities that are combinations of the xy and yxcouplings may also be computed from the xx and/or yy voltagemeasurements. For example, a compensated quantity proportional to thesum of the xy and yx couplings may be computed from the xx and/or yyvoltage measurements as follows:

$\begin{matrix}{{{CXYplusYX}_{ijxx} = {{- \frac{V_{SHS\_ ijxx}}{V_{DC\_ ijxx}}} = \frac{\left( {Z_{ijxy} + Z_{ijyx}} \right)}{\left( {Z_{ijxx} + z_{ijyy}} \right)}}}{{CXYplusYX}_{ijyy} = {{- \frac{V_{SHS\_ ijyy}}{V_{DC\_ ijyy}}} = \frac{\left( {Z_{ijxy} + Z_{ijyx}} \right)}{\left( {Z_{ijxx} + Z_{ijyy}} \right)}}}} & (35)\end{matrix}$

where CXYplusYX_(ijxx) and CXYplusYX_(ijyy) represent gain compensatedquantities computed from the xx and yy voltage measurements. Since thefirst and second quantities in each of Equations 32, 33, 34, and 35 areidentical they may be combined (e.g., averaged) to improve the signal tonoise ratio as described above with respect to the quantities inEquations 30 and 31.

Gain compensated quantities that are combinations of the xy and yxcouplings may also be computed from the xx, yy, xy, and yx voltagemeasurements. For example, a compensated quantity proportional to thedifference between the xy and yx couplings may be computed from the xx,yy, xy, and yx voltage measurements as follows:

$\begin{matrix}{{CXYminusYX}_{ij} = {\sqrt{- \frac{V_{DC\_ ijxy}V_{DC\_ ijyx}}{V_{DC\_ ijxx}V_{DC\_ ijyy}}} = \frac{\left( {Z_{ijxy} - Z_{ijyx}} \right)}{\left( {Z_{ijxx} + Z_{ijyy}} \right)}}} & (36)\end{matrix}$

where CXYminusYX_(ij) represents the gain compensated quantity. Acompensated quantity proportional to the difference between the xy andyx couplings may be computed, for example, as follows:

$\begin{matrix}{{CXYminusYX} = {\sqrt{- \frac{V_{{DC\_}12{xy}}V_{{DC\_}21{yx}}}{V_{{DC\_}11{xx}}V_{{DC\_}22{yy}}}} = \sqrt{\frac{\left( {Z_{12{xy}} - Z_{12{yx}}} \right)}{\left( {Z_{11{xx}} + Z_{11{yy}}} \right)}\frac{\left( {Z_{21{xy}} - Z_{21{yx}}} \right)}{\left( {Z_{22{xx}} + Z_{22{yy}}} \right)}}}} & (37)\end{matrix}$

The gain compensated quantities in Equations 35 and 36 may be combinedto obtain gain compensated xy and yx quantities CXY_(ij) and CYX_(ij),for example, as follows:

$\begin{matrix}{{{CXY}_{ij} = {\frac{{CXYplusYX}_{ijxx} + {CXYminusYX}_{ij}}{2} = \frac{Z_{ijxy}}{\left( {Z_{ijxx} + Z_{ijyy}} \right)}}}{{CXY}_{ij} = {\frac{{CXYplusYX}_{ijyy} + {CXYminusYX}_{ij}}{2} = \frac{Z_{ijxy}}{\left( {Z_{ijxx} + Z_{ijyy}} \right)}}}{{CYX}_{ij} = {\frac{{CXYplusYX}_{ijxx} - {CXYminusYX}_{ij}}{2} = \frac{Z_{ijyx}}{\left( {Z_{ijxx} + Z_{ijyy}} \right)}}}{{CYX}_{ij} = {\frac{{CXYplusYX}_{ijyy} - {CXYminusYX}_{ij}}{2} = \frac{Z_{ijyx}}{\left( {Z_{ijxx} + Z_{ijyy}} \right)}}}} & (38)\end{matrix}$

As described above with respect to Equation 19, a phase shift andattenuation may be computed for the compensated quantities listed above,for example, as follows:

$\begin{matrix}{{{PS}\overset{def}{=}{\frac{180}{~\pi}{{angle}\left( {1 + {C\; Q}} \right)}}}{{AT}\overset{def}{=}{20\; \log \; 10\left( {1 + {C\; Q}} \right)}}} & (39)\end{matrix}$

where PS represents the phase shift, AT represents attenuation, and CQrepresents the compensated quantity (e.g., one of the quantitiescomputed in Equations 30-38).

With respect to the embodiment depicted on FIG. 11D and Equation 37, again compensated measurement CXYminusYX sensitive to xy-yx may beobtained, for example, as follows:

$\begin{matrix}{{CXYminusYX} = {\sqrt{\frac{V_{{DC\_}13{xy}}V_{{{DC}\_}41{yx}}}{V_{{{DC}\_}11{xx}}V_{{{DC}\_}43{yy}}}} = \sqrt{\frac{\left( {Z_{13{xy}} - Z_{13{yx}}} \right)}{\left( {Z_{11{xx}} + Z_{11{yy}}} \right)}\frac{\left( {Z_{42{xy}} - Z_{42{yx}}} \right)}{\left( {Z_{43{xx}} + Z_{43{yy}}} \right)}}}} & (40)\end{matrix}$

Likewise, a gain compensated measurement CXXminusYY sensitive to xx-yymay be obtained, for example, as follows:

$\begin{matrix}{{CXXminusYY} = {\sqrt{\frac{V_{{{SHC}\_}13{xy}}V_{{{SHC}\_}41{yx}}}{V_{{{DC}\_}11{xx}}V_{{{DC}\_}43{yy}}}} = \sqrt{\frac{\left( {Z_{13{xx}} - Z_{13{yy}}} \right)}{\left( {Z_{11{xx}} + Z_{11{yy}}} \right)}\frac{\left( {Z_{42{xx}} - Z_{42{yy}}} \right)}{\left( {Z_{43{xx}} + Z_{43{yy}}} \right)}}}} & (41)\end{matrix}$

Gain Compensated Axial Term

Techniques are disclosed above for obtaining fully gain compensatedquantities related to each of the eight non-axial three-dimensionalimpedances (i.e., the xz, zx, yz, zy, xx, yy, xy, and yx terms). A gaincompensated zz (axial) coupling may be also obtained from the DCharmonic voltage coefficients, for example, as follows:

$\begin{matrix}{{CZZ} = {\sqrt{\frac{V_{{{DC}\_}12{zz}}V_{{{DC}\_}21{zz}}}{V_{{{DC}\_}11{zz}}V_{{{DC}\_}22{zz}}}} = \sqrt{\frac{Z_{12{zz}}Z_{21{zz}}}{Z_{11{zz}}Z_{22{zz}}}}}} & (42)\end{matrix}$

where CZZ represents the compensated measurement (the zz direct couplingimpedance). A phase shift and attenuation for CZZ may also be computed.

Gain Compensated Axial Cross Terms Using Tilted Moments

FIG. 13 depicts the antenna moments for an electromagnetic logging toolincluding tilted transmitters T1 and T2. The transmitters are deployedabout axially spaced receivers R1 and R2, each of which includescollocated axial and transverse antennas R_(1z), R_(1x) and R_(2z),R_(2x). The measurement tool depicted on FIG. 13 may be used to obtaingain compensated measurements, for example, as described above withrespect to FIG. 4 and Equations 10-28. For example, with respect toEquation 12, the DC and first harmonic cosine voltage coefficients maybe expressed as follows:

$\begin{matrix}{{V_{DC\_ xx} = {_{T\; 2}_{R\; 2x}{\sin \left( \beta_{T\; 2} \right)}\frac{Z_{xx} + Z_{yy}}{2}}}{V_{DC\_ zz} = {_{T\; 1}_{R\; 1z}{\cos \left( \beta_{T\; 1} \right)}Z_{zz}}}{V_{FHC\_ xz} = {_{T\; 2}_{Rz}{\sin \left( \beta_{T\; 2} \right)}Z_{xz}}}{V_{FHC\_ zx} = {_{T\; 1}_{Rx}{\cos \left( \beta_{T\; 1} \right)}Z_{zx}}}} & (43)\end{matrix}$

where β_(T1) represents the tilt angle between the T1 antenna moment andthe axis of the electromagnetic measurement tool. Notice thatsin(β_(T2)) and cos(β_(T1)) may be lumped with the transmitter gainssuch that g_(T2x)=g_(T2) sin(β_(T2)) and g_(T1z)=g_(T1) cos(β_(T1)). Thesine and cosine terms thus cancel in computing the aforementioned ratiosin the same way that the transmitter gains cancel. In this way any ofthe compensated quantities described above with respect to Equations10-28 may be computed using a measurement tool including tiltedtransmitters as depicted on FIG. 13.

It will be understood that the various methods disclosed herein forobtaining fully gain compensated quantities may be implemented on a on adownhole processor. By downhole processor it is meant an electronicprocessor (e.g., a microprocessor or digital controller) deployed in thedrill string (e.g., in the electromagnetic logging tool or elsewhere inthe BHA). In such embodiments, the fully gain compensated quantities maybe stored in downhole memory and/or transmitted to the surface whiledrilling via known telemetry techniques (e.g., mud pulse telemetry orwired drill pipe). Alternatively, the harmonic fitting coefficients maybe transmitted uphole and the compensated quantities may be computed atthe surface using a surface processor. Whether transmitted to thesurface or computed at the surface, the quantity may be utilized in aninversion process (along with a formation model) to obtain variousformation parameters as described above.

Although gain compensated directional propagation measurements have beendescribed in detail, it should be understood that various changes,substitutions and alternations can be made herein without departing fromthe spirit and scope of the disclosure as defined by the appendedclaims.

What is claimed is:
 1. A method for making downhole electromagneticlogging while drilling measurements, the method comprising (a) rotatingan electromagnetic logging while drilling tool in a subterraneanwellbore, the logging tool including at least one axial transmitterantenna, at least one transverse transmitter antenna, at least one axialreceiver antenna, and at least one transverse receiver antenna; (b)acquiring a plurality of electromagnetic voltage measurements from theaxial and transverse receiver antennas while rotating in (a); (c)processing the voltage measurements acquired in (b) to compute harmonicvoltage coefficients; (d) processing a ratio of selected ones of theharmonic voltage coefficients to compute at least one gain compensatedquantity including an axial cross term.
 2. The method of claim 1,wherein the axial cross term is an xz coupling, a zx coupling, a yzcoupling, a zy coupling, or a combination thereof.
 3. The method ofclaim 1, wherein the processing in (d) is performed by a downholeprocessor.
 4. The method of claim 3, further comprising: (e)transmitting the gain compensated quantity to a surface location; and(f) causing a surface computer to invert the gain compensated quantityto obtain one or more properties of a subterranean formation.
 5. Themethod of claim 1, further comprising: (e) processing the gaincompensated quantity to compute a gain compensated phase shift and again compensated attenuation.
 6. The method of claim 1, wherein theharmonic voltage coefficients computed in (c) comprise DC, firstharmonic sine, first harmonic cosine, second harmonic sine, and secondharmonic cosine voltage coefficients.
 7. The method of claim 1, whereinthe gain compensated quantity computed in (d) comprises at least onequantity proportional to an xz coupling, a zx coupling, a yz coupling,or a zy coupling.
 8. The method of claim 7, wherein the gain compensatedquantity is computed using at least one of the following mathematicalequations:${CZX} = \sqrt{\frac{V_{{{FHC}\_}12{zx}} \cdot V_{{{FHC}\_}21{xz}}}{V_{{{DC}\_}22{xx}} \cdot V_{{{DC}\_}11{zz}}}}$${CZX} = \sqrt{\frac{V_{{{FHC}\_}11{zx}} \cdot V_{{{FHC}\_}22{xz}}}{V_{{{DC}\_}21{xx}} \cdot V_{{{DC}\_}12{zz}}}}$${CXZ} = \sqrt{\frac{V_{{{FHC}\_}21{zx}} \cdot V_{{{FHC}\_}12{xz}}}{V_{{{DC}\_}11{xx}} \cdot V_{{{DC}\_}22{zz}}}}$${CXZ} = \sqrt{\frac{V_{{{FHC}\_}22{zx}} \cdot V_{{{FHC}\_}11{xz}}}{V_{{{DC}\_}12{xx}} \cdot V_{{{DC}\_}21{zz}}}}$wherein CZX represents the quantity proportional to the zx coupling, CXZrepresents the quantity proportional to the xz coupling, V_(DC) _(—)_(11xx), V_(DC) _(—) _(12xx), V_(DC) _(—) _(21xx), V_(DC) _(—) _(22xx),V_(DC) _(—) _(11zz), V_(DC) _(—) _(12zz), V_(DC) _(—) _(21zz), andV_(DC) _(—) _(22zz) represent the DC voltage coefficients, and V_(FHC)_(—) _(11zx), V_(FHC) _(—) _(12zx), V_(FHC) _(—) _(21zx), V_(FHC) _(—)_(22zx), V_(FHC) _(—) _(11zx), V_(FHC) _(—) _(12zx), V_(FHC) _(—)_(21zx), and V_(FHC) _(—) _(22zx) represent the first harmonic cosinevoltage coefficients.
 9. The method of claim 7, wherein the gaincompensated quantity is computed using at least one of the followingmathematical equations:${CZY} = \sqrt{\frac{V_{{{FHS}\_}12{zx}} \cdot V_{{{FHS}\_}21{xz}}}{V_{{{DC}\_}22{xx}} \cdot V_{{{DC}\_}11{zz}}}}$${CZY} = \sqrt{\frac{V_{{{FHS}\_}11{zx}} \cdot V_{{{FHS}\_}22{xz}}}{V_{{{DC}\_}21{xx}} \cdot V_{{{DC}\_}12{zz}}}}$${CYZ} = \sqrt{\frac{V_{{{FHS}\_}21{zx}} \cdot V_{{{FHS}\_}12{xz}}}{V_{{{DC}\_}11{xx}} \cdot V_{{{DC}\_}22{zz}}}}$${CYZ} = \sqrt{\frac{V_{{{FHS}\_}11{zx}} \cdot V_{{{FHS}\_}22{xz}}}{V_{{{DC}\_}12{xx}} \cdot V_{{{DC}\_}21{zz}}}}$wherein CZY represents the quantity proportional to the zy coupling, CYZrepresents the quantity proportional to the yz coupling, V_(DC) _(—)_(11xx), V_(DC) _(—) _(12xx), V_(DC) _(—) _(21xx), V_(DC) _(—) _(22xx),V_(DC) _(—) _(11zz), V_(DC) _(—) _(12zz), V_(DC) _(—) _(21zz), andV_(DC) _(—) _(22zz) represent the DC voltage coefficients, and V_(FHS)_(—) _(11zx), V_(FHS) _(—) _(12zx), V_(FHS) _(—) _(21zx), V_(FHS) _(—)_(22zx), V_(FHS) _(—) _(11zx), V_(FHS) _(—) _(12zx), V_(FHS) _(—)_(21zx), and V_(FHS) _(—) _(22zx) represent the first harmonic sinevoltage coefficients.
 10. The method of claim 1, wherein the gaincompensated quantity computed in (d) comprises at least one quantityproportional to a product of an xz coupling and a zx coupling or aproduct of a yz coupling and a zy coupling.
 11. The method of claim 10,wherein the gain compensated quantity is computed using at least one ofthe following mathematical equations:${CXZZX} = \sqrt{\frac{V_{FHC\_ ijzx} \cdot V_{FHC\_ ijxz}}{V_{DC\_ ijxx} \cdot V_{DC\_ ijzz}}}$${CXZZX} = \sqrt{\frac{V_{FHC\_ ijzx} \cdot V_{FHC\_ iixz}}{V_{DC\_ ijxx} \cdot V_{DC\_ ijiz}}}$${CXZZX} = \sqrt{\frac{V_{{FHC}\_ {iizx}} \cdot V_{{FHC}{\_ jixz}}}{V_{{DC}{\_ jixx}} \cdot V_{{DC}{\_ iizz}}}}$wherein CXZZX represents quantity proportional to the a product of thexz coupling and the zx coupling, V_(DC) _(—) _(ijxx), V_(DC) _(—)_(jixx), V_(DC) _(—) _(iizz), and V_(DC) _(—) _(ijzz), represent the DCvoltage coefficients, and V_(FHC) _(—) _(iizx), V_(FHC) _(—) _(iixz),V_(FHC) _(—) _(ijzx), V_(FHC) _(—) _(ijxz), and V_(FHC) _(—) _(jixz)represent the first harmonic cosine voltage coefficients.
 12. The methodof claim 10, wherein the gain compensated quantity is computed using atleast one of the following mathematical equations:${CYZZY} = \sqrt{\frac{V_{FHS\_ ijzx} \cdot V_{FHS\_ ijxz}}{V_{DC\_ ijxx} \cdot V_{DC\_ ijzz}}}$${CYZZY} = \sqrt{\frac{V_{FHS\_ ijzx} \cdot V_{FHS\_ iixz}}{V_{DC\_ ijxx} \cdot V_{DC\_ iizz}}}$${CYZZY} = \sqrt{\frac{V_{FHS\_ iizx} \cdot V_{FHS\_ jixz}}{V_{DC\_ jixx} \cdot V_{DC\_ iizz}}}$wherein CYZZY represents the quantity proportional sensitive to theproduct of the yz coupling and the zy coupling, V_(DC) _(—) _(ijxx),V_(DC) _(—) _(jixx), V_(DC) _(—) _(iizz), and V_(DC) _(—) _(ijzz),represent the DC voltage coefficients, and V_(FHS) _(—) _(iizx), V_(FHS)_(—) _(iixz), V_(FHS) _(—) _(ijzx), V_(FHS) _(—) _(ijxz), and V_(FHS)_(—) _(jixz) represent the first harmonic sine voltage coefficients. 13.The method of claim 1, wherein processing the ratio in (d) comprisescomputing a ratio of a first product of first harmonic voltage cosinecoefficients to a second product of DC voltage coefficients to computethe gain compensated quantity.
 14. The method of claim 13, wherein theratio of the first product to the second product is expressedmathematically as follows:${CR} = \frac{V_{FHC\_ zx} \cdot V_{FHC\_ xz}}{V_{DC\_ xx} \cdot V_{DC\_ zz}}$wherein CR represents the ratio, V_(FHC) _(—) _(zx) and V_(FHC) _(—)_(xz) represent the first harmonic cosine coefficients, and V_(DC) _(—)_(xx) and V_(DC) _(—) _(zz) represent the DC voltage coefficients. 15.The method of claim 1, wherein processing the ratio in (d) comprisescomputing a ratio of a first product of first harmonic voltage sinecoefficients to a second product of DC voltage coefficients to computethe gain compensated quantity.
 16. The method of claim 15, wherein theratio of the first product to the second product is expressedmathematically as follows:${CR} = \frac{V_{FHS\_ zx} \cdot V_{FHS\_ xz}}{V_{DC\_ xx} \cdot V_{DC\_ zz}}$wherein CRyz represents the ratio, V_(FHS) _(—) _(zx) and V_(FHS) _(—)_(xz) represent the first harmonic sine coefficients, and V_(DC) _(—)_(xx) and V_(DC) _(—) _(zz) represent the DC voltage coefficients. 17.The method of claim 1, wherein (d) further comprises computing a squareroot of the ratio.
 18. A method for making downhole electromagneticlogging while drilling measurements, the method comprising (a) rotatingan electromagnetic logging while drilling tool in a subterraneanwellbore, the logging tool including at least one axial transmitterantenna, at least one transverse transmitter antenna, at least one axialreceiver antenna and at least one transverse receiver antenna. (b)causing the axial transmitter antenna to transmit an electromagneticwave while rotating in (a); (c) acquiring electromagnetic voltagemeasurements from the axial and transverse receiver antennas while theaxial transmitter antenna is transmitting in (b); (d) causing thetransverse transmitter antenna to transmit an electromagnetic wave whilerotating in (a); (e) acquiring electromagnetic voltage measurements fromthe axial and transverse receiver antennas while the transversetransmitter antenna is transmitting in (d); (f) processing the voltagemeasurements acquired in (b) and (d) to compute corresponding harmonicvoltage coefficients; (g) processing a ratio of selected ones of theharmonic voltage coefficients computed in (f) to compute at least onegain compensated quantity including an axial cross term.
 19. Anelectromagnetic logging while drilling tool comprising: a logging whiledrilling tool body; an axial transmitter antenna and a transversetransmitter antenna deployed on the tool body; an axial receiver antennaand a transverse receiver antenna deployed on the tool body; acontroller configured to (i) cause the axial transmitter antenna and thetransverse transmitter antenna to transmit corresponding electromagneticwaves; (ii) acquire electromagnetic voltage measurements from the axialand transverse receiver antennas while the axial transmitter antenna andthe transverse transmitter antenna are transmitting; (iii) process theelectromagnetic voltage measurements to compute harmonic voltagecoefficients; and (iv) process a ratio of selected ones of the harmonicvoltage coefficients to compute at least one gain compensated quantityincluding an axial cross term.
 20. The logging while drilling tool ofclaim 19, wherein the axial transmitter antenna and the transversetransmitter antenna are collocated with one another.
 21. The loggingwhile drilling tool of claim 19, wherein the axial receiver antenna andthe transverse receiver antenna are collocated with one another.
 22. Thelogging while drilling tool of claim 19, wherein the axial receiverantenna and the transverse receiver antenna are non-collocated and aredeployed axially between the axial transmitter antenna and thetransverse transmitter antenna.